Isotropic sample

In an isotropic sample, where the C—H bond directions are uniformely destributed in space, the resulting powder line shape is the famous Pake spectrum of the I = 1 spin system depicted for convenience in Fig. 1. 

Birefringence is one of the simplest methods for the characterization of molecular orientation in polymers. The polarizability of a structural unit is usually not equivalent in all directions, leading to three independent refractive indices along its principal axes. In an isotropic sample, a single averaged macroscopic refractive index is observed whereas birefringence or trirefringence is observed... 

Ap and As are the absorbances measured with p- and s-polarization, respectively, and A0 — (Ap + 2As)/3 is the structural absorbance spectrum that would be measured for an isotropic sample. The order parameter of the main chain can be determined using the Legendre addition theorem (Equation (24)). 

Consequently, the measurement of three spectra and the determination of a from the isotropic sample allow calculation of (P2) and (P4). This method has been applied successfully to PET [53] and rubber networks [52], and to study the... 

The measurement of four independent spectra then leads to the complete resolution of the system, assuming that the depolarization ratio is the same for the oriented and isotropic samples. 

Silk fibers, a basic system with a uniaxial symmetry, have also been investigated by Raman spectromicroscopy [63] that is one of the rare techniques capable of providing molecular data on such small (3-10 pm diameter) single filaments. The amide I band of the silk proteins has been particularly studied to determine the molecular orientation using the cylindrical Raman tensor approximation. In this work, it was assumed that Co Ci, C2 and the a parameter was determined from an isotropic sample using the following expression of the depolarization ratio... 

If several isotropic samples from the same material are studied, Xc arranges them in the order of their crystallinity - but without telling the correct value. 

Classical Evaluation of Porod s Law. The practical evaluation of Porod s law in the most common case of an isotropic sample measured with point-focus is demonstrated in Fig. 8.11. By variation of the fluctuation background a long and linear Porod region can be received. Nevertheless, the line still shows a negative slope (Fig. 8.11). Ruland s theory of the systematic deviations from Porod s law [132] explains this finding. 

Figure 1. Polarized XAS. The orientations of i and k and the angle 6 ( see Equation 2) are shown for a typical PXAS experiment. For this orientation, there can be no Cu-S contribution to the EXAFS (O = 90°), The Cu-N contribution to the EXAFS will be 3cos20-times that observed for an isotropic sample.

In isotropic samples yielding begins in the domains with an angle 0=7t/4 and Eq. 13 reduces to... 

Isotropic samples will have no effect on the polarized light no matter how the crystal is oriented, since all crystal axes are completely equivalent. This effect is known as complete or isotropic extinction (Fig. 3a). Noncrystalline, amorphous samples will exhibit the same effect. 

The behaviour of chain-length distribution is non-characteristic during thermal-oxidative degradation of isotropic PP films the degradation shifts towards decrease in molecular mass, while during oxidation in deformed films it shifts towards increase in molecular mass. It means that with oxidation in isotropic samples in the induction period the destruction of molecules prevails. On the other hand, with oxidation in deformed samples attachement and cross-linking dominate. 

The refractive index is an important quantity for characterizing the structure of polymers. This is because it depends sensitively on the chemical composition, on the tacticity, and - for oligomeric samples - also on the molecular weight of a macromolecular substance. The refractive indices (determined using the sodium D line) of many polymers are collected in the literature. In order to characterize a molecule s constitution one requires knowledge of the mole refraction, Rg. For isotropic samples, it can be calculated in good approximation by the Lorentz-Lorenz equation ... 

The two principle modes of the magnetostriction (A -° and As or Xy-2) introduced above are illustrated in fig. 2. With respect to the non-magnetic fictitious state, a spherical, isotropic sample exhibits a relative volume change AV/V = A" 0, when it becomes magnetic. In addition, when one forces the moments to be directed along an applied magnetic field B, an anisotropic deformation is induced, which transfers the sphere into an ellipsoid with the same volume. 

Since in the elastic region (0 In L/0 In Ot.p is always positive, the force-temperature coefficient at constant length and pressure must be of opposite sign to PL. For rubbers, at some extensions the isometric inversion [(0 In f/6T), P = 0] must occur since pL of the isotropic sample is always positive. For solids, such measurements correspond to the determination of the coefficient (5 of an elastically stretched sample which, however, does not differ from the usual coefficient of thermal expansion. 

Fig. 14. Heat (1, 1 ) and heat to work ratio (2, 2 ) for reversible deformation of LDPE at 20 °C 7 81. 1,2 — isotropic sample 1, 2 — cold-drawn sample (A. = 4, deformation along orientation axis). Dotted curves correspond to the equation Q = fiTEe and solid curve 2 to the equation Q/W = 2PT/c with P = 2.4 x 10"4 K 1 and E = 0.1 GPa...

Irradiation procedures. Mesophase solutions and neat solid samples of BN were prepared and sealed under N2 or vacuum in Kimax capillary tubes. Isotropic samples were either degassed (freeze-pump-thaw techniques) and sealed in pyrex tubes or saturated with N2 in pyrex tubes. Nitrogen was bubbled through the latter solutions during irradiation periods. When ther-mostatted, samples were placed in a temperature controlled ( 1°) water bath. All samples were irradiated with a 450 W Hanovia medium pressure Hg arc and were stored at -30°C until their futher use. Usually, a "dark sample was prepared and treated in an identical fashion to the irradiated samples except that it was shielded from the light. JSN from each tube was recovered by either column chromatography (silica or alumina and pentane eluant) at 4°C followed by solvent removal at 0°C and reduced pressure or by hplc (tr-hexane) at room temperature followed by solvent removal at 0°C and reduced pressure. Neat solid samples were dissolved in one of either benzene, tetrahydrofuran or toluene and were frozen until analyzed. 

This contribution is purely anisotropic, and thus vanishes upon orientational averaging and does not contribute in the case of isotropic samples such as a liquid. 

The expressions for Raman and ROA intensities depend on products of the various transition tensors. For ROA, isotropic samples are of interest and rotational averages of the products of these tensors are therefore required. The rotational averages can be expressed by double contractions of the isotropic (is), the anisotropic (anis), and the antisymmetric (a) part, which for a second rank tensor T are defined as... 

Another way to disentangle linear polyethylenes, and thus control the interphase without using a solvent, is to anneal the polymer in the hexagonal phase. Bassett has discussed the role of the hexagonal phase in the crystallization of polyethylene extensively in an earlier chapter in this book Briefly, polyethylene exhibits a number of different crystal structures, with the hexagonal phase being observed in linear polyethylenes at elevated pres-sure/temperature in isotropic samples or at ambient pressure in oriented samples. For this reason, we have to distinguish between these two situations, namely isotropic and oriented polyethylene. However, we will focus only on isotropic polyethylene and will refer readers to reference [18,19] for an overview of oriented polyethylene. 

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